Can anybody give me (an) example(s) of a symbolic dynamical system (preferably arising from a substitution) which has a pair of points which are proximal but not asymptotic? I would prefer to work with the space of two-sided infinite words $\mathcal{A}^\mathbb{Z}$, since the shift operator $\sigma$ there is invertible.
By a proximal pair $x,y$, I mean that for every $\epsilon>0$ there is an $n\in\mathbb{Z}$ such that $d(\sigma^n(x),\sigma^n(y))<\epsilon$, where $d$ is the usual metric on $\mathcal{A}^\mathbb{Z}$.
By an asymptotic pair $x,y$, I mean that for every $\epsilon>0$ there is an $N\in\mathbb{N}$ such that for all $n\in\mathbb{Z}$, $|n|>N$ implies that $d(\sigma^n(x),\sigma^n(y))<\epsilon$.
An example with (the negation of) a 'weaker' version of asymptoticity, meaning that there is an $N\in\mathbb{N}$ such that either for all $n\in\mathbb{Z}$, $n>N$ implies $d(\sigma^n(x),\sigma^n(y))<\epsilon$, or for all $n\in\mathbb{Z}$, $n<-N$ implies $d(\sigma^n(x),\sigma^n(y))<\epsilon$, is also welcome.
Any help or reference will also be appreciated.
A (non-minimal) example is given the full shift $A^\mathbf{Z}$ with $A=\{0,1\}$, $x_n=0$ for all $n$, $y_{\pm n^2}=1$, $y_n=0$ for other $n$, so $x$ and $y$ are proximal but not asymptotic. (Alternatively, we can restrict to the orbit closure of $y$.)
Here's now a minimal example. It's a bit more complicated but is very classical in topological dynamics.
If $Y$ is a subset of the circle $C=\mathbf{R}/\mathbf{Z}$, let $X=C_Y$ denote the set which informally is obtained from the circle by doubling all points in $Y$; formally, say $C_Y=C\times\{0\}\cup Y\times\{1\}$, and denote the elements $(c,0)$ as $c=c^+=c^-$ if $c\notin Y$, as $c^-$ if $c\in Y$, and denote $(c,1)$ as $c^+$. Endow $C_Y$ with the obvious cyclic ordering (so $a^\pm<b^\pm<c^\pm$ whenever $a<b<c$, and $a^-<a^+<b$ whenever $a\neq b$), and the corresponding topology. This is a compact topology.
Now let $t$ be an irrational, and denote $(t)=\mathbf{Z}t\subset C$. So the self-homeomorphism $\sigma:x\mapsto x+t$ on $C$ lifts to a self-homeomorphism of $C_{(t)}$. Then the pair $(0^-,0^+)$ is proximal but not asymptotic. (Indeed, if $|n_i|\to\infty$ and $n_it$ converges to an element $y$ of $C$, then if $y\notin (t)$, $\sigma^{n_i}(0^\pm)\to y$, while if $y\in (t)$ then $\sigma^{n_i}(0^\pm)\to y^\pm$ so $\sigma^{n_i}(0^+)$ and $\sigma^{n_i}(0^-)$ have distinct limits.)
Added: this is symbolic. Indeed you can embed it as a subshift on two letters $\{0,1\}$ by mapping a point $x$ to the sequence $(x_n)$, where $x_n=1$ if and only if $\sigma^n(x)\in [0^+,t^-]$ (where $t$ is chosen with $0<t<1$).